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%I #17 Jan 23 2024 15:12:14
%S 1,2,20,1600,1408000,17346560000,3633063526400000,
%T 15218176499384320000000,1466155647574283911168000000000,
%U 3672576800382377947366110003200000000000,266783946802402043703868836144710942720000000000000
%N a(n) = Product_{k=0..n} (3*k+2)^(n-k).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Glaisher-KinkelinConstant.html">Glaisher-Kinkelin Constant</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PolygammaFunction.html">Polygamma Function</a>
%F a(n) ~ A^(1/3) * 2^(n/2 + 1/3) * 3^(n^2/2 + n/2 - 1/72) * Pi^(n/2 + 1/3) * n^(n^2/2 + 2*n/3 + 5/36) / (Gamma(2/3)^(n + 2/3) * exp(3*n^2/4 + 2*n/3 - Pi/(18*sqrt(3)) + PolyGamma(1, 1/3) / (12*sqrt(3)*Pi) + 1/36)), where A = A074962 is the Glaisher-Kinkelin constant and PolyGamma(1, 1/3) = 10.095597125427094081792004... (PolyGamma[1, 1/3] in Mathematica or Psi(1, 1/3) in Maple).
%F PolyGamma(1, 1/3) = 3^(3/2) * A261024 + 2*Pi^2/3.
%F a(n) = 3^(n*(n+1)/2) * BarnesG(n + 5/3) / (BarnesG(2/3) * Gamma(2/3)^(n+1)). - _Vaclav Kotesovec_, Jan 23 2024
%t Table[Product[(3*k+2)^(n-k),{k,0,n}],{n,0,12}]
%t (* or *)
%t Table[1/FullSimplify[Gamma[2/3]^((v-2)/3) * 3^((v-2)/18) * Exp[Integrate[(E^((3-v)*x) - E^x)/(x*(E^(3*x)-1)^2) + (v-2) * (1/(3*x*(E^(3*x)-1)) + 1/(6*x*E^(3*x)) - (v+2)/(18*x*E^x)), {x, 0, Infinity}]]], {v, 2, 35, 3}]
%t Table[3^(n*(n+1)/2) * BarnesG[n + 5/3] / (BarnesG[2/3] * Gamma[2/3]^(n+1)), {n, 0, 12}] // Round (* _Vaclav Kotesovec_, Jan 23 2024 *)
%Y Cf. A263416, A263406, A263415, A369468.
%K nonn
%O 0,2
%A _Vaclav Kotesovec_, Oct 17 2015
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